{
  "id": "2408.11232",
  "version": "v1",
  "published": "2024-08-20T23:13:02.000Z",
  "updated": "2024-08-20T23:13:02.000Z",
  "title": "Large sum-free sets in finite vector spaces I",
  "authors": [
    "Christian Reiher",
    "Sofia Zotova"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $p$ be a prime number with $p\\equiv 2\\pmod{3}$ and let $n\\ge 1$ be a dimension. It is known that a sum-free subset of ${\\mathbb F}_p^n$ can have at most the size $\\frac13(p+1)p^{n-1}$ and that, up to automorphisms of ${\\mathbb F}_p^n$, the only extremal example is the `cuboid' $\\bigl[\\frac{p+1}3, \\frac{2p-1}3\\bigr]\\times {\\mathbb F}_p^{n-1}$. For $p\\ge 11$ we show that if a sum-free subset of ${\\mathbb F}_p^n$ is not contained in such an extremal one, then its size is at most $\\frac13(p-2)p^{n-1}$. This bound is optimal and we classify the extremal configurations. The remaining cases $p=2, 5$ are known to behave differently. For $p=3$ the analogous question was solved by Vsevolod Lev, and for $p\\equiv 1\\pmod{3}$ it is less interesting.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-20T23:13:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B13",
      "11B30",
      "11P70"
    ],
    "keywords": [
      "finite vector spaces",
      "large sum-free sets",
      "sum-free subset",
      "extremal configurations",
      "extremal example"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}